Spectrum of walk matrix for Koch network and its application

Abstract

Various structural and dynamical properties of a network are encoded in the eigenvalues of walk matrix describing random walks on the network. In this paper, we study the spectra of walk matrix of the Koch network, which displays the prominent scale-free and small-world features. Utilizing the particular architecture of the network, we obtain all the eigenvalues and their corresponding multiplicities. Based on the link between the eigenvalues of walk matrix and random target access time defined as the expected time for a walker going from an arbitrary node to another one selected randomly according to the steady-state distribution, we then derive an explicit solution to the random target access time for random walks on the Koch network. Finally, we corroborate our computation for the eigenvalues by enumerating spanning trees in the Koch network, using the connection governing eigenvalues and spanning trees, where a spanning tree of a network is a subgraph of the network, that is, a tree containing all the nodes.

Exactly solvable tight-binding model on two scale-free networks with identical degree distribution

“…Next, we will solve the energy spectrum of H for G 1 and G 2 by exact matrix renormalization. Similar derivation of spectrum for other renormalizable structures can be found at [37,38].…”

Effects of heterogeneity in site–site couplings for tight-binding models on scale-invariant structures

“…Notice that by tracking the preimage of the spectrum T (t) under function R one can obtain only 2N t eigenvalues of T (t+1) . Fortunately, the left N t+1 − 2N t eigenvalues are uniformly 0 that is the only singularity of R. More detailed discussion about the iterative derivation of spectrum can be found at [45,46]. So far we are able to obtain the full spectrum of T (t) by tracking the flow generated by R. As t → ∞, the spectrum grows into a Julia set J R given as…”

The normalized Laplacian spectrum of subdivisions of a graph

“…The simple relationship between the spectrum of the Markov and the normalized Laplacian matrices of a subdivision graph facilitates the calculations to obtain the exact distribution and values of all eigenvalues. One particular interesting result is that the multiplicity of the exceptional eigenvalues of σ n do not increase exponentially with n as in [28,29] but is a constant determined by the value of the circuit rank of the initial graph.…”